1. MT411 Robotic Engineering
Chapter 10
Velocity Kinematics – The Jacobian
Narong Aphiratsakun, D.Eng
Asian Institution of Technology (AIT)

2. Velocity Kinematic
To follow any prescript velocity, it is important to know the relationship between the velocity of the tool and the joint velocity.
So, we are going to find velocity relationship relating the linear and angular velocities of the end effector to the joint velocities.
The velocity relationships are determined by the Jacobian of the function. The jacobian is a matrix that generalizes the notions of the ordinary derivative of the scalar function.

3. Velocity Kinematics: 2 links manipulator
Differentiate both side, (prove is on next page)

4. Velocity Kinematics: 2 links manipulator

5. Velocity Kinematics: 2 links manipulator
Put in vector form
Rewrite as
Tool velocity
Joints velocity

6. Velocity Kinematics: 2 links manipulator
**J is called Jacobian of the manipulator

7. Velocity Kinematics: Tool and Joint Velocity
Joints velocity
Tool velocity
In this chapter, we are going to find Jacobian matrix of the manipulator.
Tool velocity
Joints velocity

8. Derivation of the Jacobian
As robot moves about, both joint variables and end effector position, d, and orientation, R, will be function of time.
The objective of this section is to relate the linear and angular velocity of the end effector to the vector of joint velocities
Rewrite d position as o position in function of time as:

9. Derivation of the Jacobian
Linear velocity and Angular velocity
Joint velocity
Body velocity
Manipulator Jacobian is a 6xn matrix where n is the number of link

10. Angular and Linear Velocity
Angular velocity:
k is a unit vector in the direction of the axis of rotation.
is a time derivative of the
Linear velocity:
r is a vector from the origin to the point.

11. Combining Linear and Angular Velocity
Linear velocity Jacobian :
Angular velocity Jacobian :

12. Angular Velocity Angular velocity for different frames:
denotes the angular velocity of frame 2 that corresponds to the changing , express relative to the coordinate system frame 1.
Thus the product of expresses this angular velocity relative to the coordinate system frame 0.
Angular velocity can be obtained from:

13. Linear Velocity Linear velocity:
This is linear velocity of the end effector that would result if were equal to one and the other were zero.

14. Jacobian Analysis : 2 links manipulator
Tool velocity
joint velocity
3 linear + 3 Revolute velocity (x,y,z)

15. Jacobian Analysis : 2 links manipulator

16. Jacobian Analysis : 2 links manipulator
2 links, n =2:

17. Jacobian Analysis : 2 links manipulator

18. Jacobian Analysis : 2 links manipulator

19. Jacobian Analysis : RRP (SCARA)
Tool velocity
joint velocity
3 linear + 3 Revolute velocity (x,y,z)

20. Jacobian Analysis : RRP (SCARA)
3 links, n =3:

21. Jacobian Analysis : RRP (SCARA)

22. Jacobian Analysis : RRP (SCARA)

23. Velocity Kinematics: Further Work
Joints velocity
Tool velocity
The joint velocities are found from the end-effector velocities via inverse Jacobian.
Our next problem is to determine what the joint velocity needed for the desired end effector velocity.
Tool velocity
Joints velocity

24. Tool Velocity
It is necessary to relate the velocity of the tool frame to the velocity of the end-effector frame.
Assume frame 6 is end-effector, therefore
Prove of this equation can be read from Robot Modeling and Control, M. W. Spong, S. Hutchinson, M. Vidyasagar pg